an unbiased die rolled n times. what is the probability that the product of the scores is an even number

As long as one of the numbers is even, the product will be even

So to have an odd product, all of the factors must be odd
to have n consecutive odds is (1/2)^n

So to have an even would be
1 - (1/2)^n

e.g.
suppose we roll it three times, n = 3
according to my formula,
prob(even) = 1 - (1/2)^3 = 7/8

cases:
EEE
EE0
EOE
OEE
E00
0E0
00E
000
The first 7 will all produce an even product
so Prob(even product) = 7/8